Bingham distribution
In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.[1] It is a generalization of the Watson distribution and a special case of the Kent and Fisher–Bingham distributions.
The Bingham distribution is widely used in paleomagnetic data analysis,[2] and has been used in the field of computer vision.[3][4][5]
Its probability density function is given by
- [math]\displaystyle{ f(\mathbf{x}\,;\,M,Z) \; dS^{n-1} = {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^{-1} \cdot \exp \left( \operatorname{tr} Z M^T \mathbf{x} \mathbf{x}^T M \right)\; dS^{n-1} }[/math]
which may also be written
- [math]\displaystyle{ f(\mathbf{x}\,;\,M,Z)\; dS^{n-1} \;=\; {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^{-1} \cdot \exp\left( \mathbf{x}^T M Z M^T \mathbf{x} \right)\; dS^{n-1} }[/math]
where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and [math]\displaystyle{ {}_{1}F_{1}(\cdot;\cdot,\cdot) }[/math] is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that underlies the Bingham distribution.
See also
References
- ↑ Bingham, Ch. (1974) "An antipodally symmetric distribution on the sphere". Annals of Statistics, 2(6):1201–1225.
- ↑ Onstott, T.C. (1980) "Application of the Bingham distribution function in paleomagnetic studies[yes|permanent dead link|dead link}}]". Journal of Geophysical Research, 85:1500–1510.
- ↑ S. Teller and M. Antone (2000). Automatic recovery of camera positions in Urban Scenes
- ↑ Haines, Tom S. F.; Wilson, Richard C. (2008). Computer Vision – ECCV 2008. Lecture Notes in Computer Science. 5304. Springer. pp. 780–791. doi:10.1007/978-3-540-88690-7_58. ISBN 978-3-540-88689-1. http://opus.bath.ac.uk/56745/1/2008_eccv.pdf.
- ↑ "Better robot vision: A neglected statistical tool could help robots better understand the objects in the world around them.". MIT News. October 7, 2013. http://web.mit.edu/newsoffice/2013/better-robot-vision-1007.html.
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